Coordinate Transformations with t
x = space coordinate, t = time coordinate, v = velocity, u = pace
Galilean transformation
speed: x´ = x – vt and t´ = t, pace: x´ = x – t/u and t´ = t.
Dual Galilean transformation
speed: t´ = t – x/v and x´ = x, pace: t´ = t – ux and x´ = x.
Light: c := speed of light, k := pace of light.
speed: x = ct or x/c = t and x´ = ct´, or x´/c = t´,
pace: kx = t or x = t/k and kr′ = t´ or x´ = t´/k.
Lorentz transformation
speed: γ = (1 – v2/c2)–1/2 with x´ = γ (x − vt) and t´ = γ (t – xv/c²),
pace: γ = (1 – k2/u2)–1/2 with x´ = γ (x – t/u) and t´ = γ (t – xk²/u),
which applies only if |v| < |c| or |u| > |k|.
Dual Lorentz transformation
speed: λ = (1 − c2/v2)–1/2 with t´ = λ (t − x/v) and x´ = λ (x − t (c2/v)),
pace: λ = (1 – u2/k2)–1/2 with t´ = λ (t – ux) and x´ = λ (x – t (u/k²)),
which applies only if |v| > |c| or |u| < |k|.
Galileo-Lorentz limit
speed: c1 → ∞: γ1 → 1, t´ → t, c2 → c/2: γ2 → (1 – 4v²/c²)−1/2 and t´ → γ2 (t – 4x (v/c²)).
pace: k1 → 0: γ1 → 1, t´ → t, ç2 → 2/k: γ2 → (1 – (1/4) k²/u²)−1/2 and t′ → γ2 (t – (x/4) (k²/u)).
Dual Galileo-Lorentz limit
speed: c1 → 0: λ1 → 1, x´ → x, c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and x´ → λ2 (x – (t/4) (c²/v)).
pace: k1 → ∞: λ1 → 1, x´ → x, k2 → k/2: λ2 → (1 – 4u²/k²)−1/2 and x´ → λ2 (x – 4t (u/k²)).
If |v| = |c|, then x′ = x and t′ = t.
Coordinate Transformations with τ
x = space coordinate, t = time coordinate, v = velocity, u = pace
c := speed of light, k := pace of light.
τ := ct τ := t/k
Galilean transformation
speed: x´ = x – τ (v/c) and τ´ = τ, pace: x´ = x – τ (k/u) and τ´ = τ.
Dual Galilean transformation
speed: τ´ = τ – x (c/v) and x´ = x, pace: τ´ = τ – x (u/k) and x´ = x.
Light: x = τ and x´ = τ´
Lorentz transformation
speed: γ = (1 – v2/c2)–1/2 with x´ = γ (x − τ (v/c)) and τ´ = γ (τ – x (v/c)),
pace: γ = (1 – k2/u2)–1/2 with x´ = γ (x – τ (k/u)) and τ´ = γ (τ – x (k/u)),
which applies only if |v| < |c| or |u| > |k|.
Dual Lorentz transformation
speed: λ = (1 − c2/v2)–1/2 with τ′ = λ (τ – x (c/v)) and x´ = λ (x − τ (c/v)),
pace: λ = (1 – u2/k2)–1/2 with τ′ = λ (τ – x (u/k)) and x´ = λ (x – τ (u/k)),
which applies only if |v| > |c| or |u| < |k|.
Galileo-Lorentz limit
speed: c1 → ∞: γ1 → 1, τ´ → τ, c2 → c/2: γ2 → (1 – 4v²/c²)−1/2 and τ′ → γ2 (τ – 4x (v/c)).
pace: ç1 → 0: γ1 → 1, τ´ → τ, ç2 → 2/k: γ2 → (1 – (1/4) k²/u²)−1/2 and τ′ → γ2 (τ – (x/4) (k/u)).
Dual Galileo-Lorentz limit
speed: c1 → 0: λ1 → 1, x´ → x, c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and x´ → λ2 (x – (τ/4) (c/v)).
pace: ç1 → ∞: λ1 → 1, x´ → x, ç2 → k/2: λ2 → (1 – 4u²/k²)−1/2 and x´ → λ2 (x – 4τ (u/k)).
If |v| = |c|, then x′ = x and τ′ = τ.
Coordinate Transformations with c := 1
x = space coordinate, t = time coordinate, v = velocity, u = pace
Galilean transformation
speed: x´ = x – vt and t´ = t, pace: x´ = x – t/u and t´ = t.
Dual Galilean transformation
speed: t´ = t – x/v and x´ = x, pace: t´ = t – ux and x´ = x.
Light: 1 = c := speed of light, 1 = k := pace of light.
speed: x = t and x´ = t´
pace: x = t and x´ = t´.
Lorentz transformation
speed: γ = (1 – v2)–1/2 with x´ = γ (x − vt) and t´ = γ (t – xv),
pace: γ = (1 – 1/u2)–1/2 with x´ = γ (x – t/u) and t´ = γ (t – x/u),
which applies only if |v| < 1 or |u| > 1.
Dual Lorentz transformation
speed: λ = (1 − 1/v2)–1/2 with t´ = λ (t − x/v) and x´ = λ (x − t (1/v)),
pace: λ = (1 – u2)–1/2 with t′ = λ (t – ux) and x´ = λ (x – tu),
which applies only if |v| > 1 or |u| < 1.
If |v| = 1, then x´ = x and t´ = t.